Chapter 2: Literature Review
2.4 Mathematical Problem Solving
2.4.7 Mathematical Modeling Process
The problem-solving and modeling process has been extensively studied by many researchers (Blum, 2002, 2015; Schoenfeld, 1992, 2007, 2014). These studies focus on the processes and mechanisms that problem-solvers perform to solve problems of a quantitative nature. Despite many definitions of modeling, fundamental
thinkers have simplified the definition to indicate the relationship that could be formed between the real world and mathematics.
Mathematical modelling is seen as “a matter of constructing an idealised, abstract model which may then be compared for its degree of similarity with a real system” (Giere, 1999, p. 50). Mathematical modeling is similar to ML in that both require applying the four main steps of Polya's model in problem-solving.
Mathematical modeling is the complete process that starts with a real-life situation that is translated into a mathematical model, and then this model must be applied to obtain the mathematical results that must be translated and validated in the original position.
(Blum & Niss, 2010).
Mathematical modeling is ‘linking classroom mathematics to something from everyday life that is not inherently mathematical’ (Cirillo, Pelesko, Felton-Koestler &
Rubel, 2016, p. 3). This means that contextual problems should be used to “elicit student thinking—to reveal bases of understanding that can be built upon”
(Schoenfeld, 1983, p. 407). Blum (2002) points to two other aspects to the definition of modeling that it has direction as it moves from reality to mathematics; and it is a
"process leading from a problem situation to a mathematical model" (p. 153).
Contextual mathematical problems are more similar to modeling problems than to problems of application (Niss, Blum & Galbraith, 2007). When it comes to using mathematics in "real-world" problems, modeling and application are sometimes used similarly. However, there is a difference between them according to Niss et al. (2007) as follows:
The term "modelling", on the one hand, tends to focus on the direction
"reality mathematics" and, on the other hand and more generally,
emphasises the processes involved. Simply put, with modelling we are standing outside mathematics looking in: "Where can I find some mathematics to help me with this problem?" In contrast, the term "application", on the one hand, tends to focus on the opposite direction "mathematics reality" and, more generally, emphasises the objects involved—in particular those parts of the real world which are (made) accessible to a mathematical treatment and to which corresponding mathematical models already exist. Again simply put, with applications we are standing inside mathematics looking out: Where can I use this particular piece of mathematical knowledge? (pp. 10-11).
What is expected of students when engaging in mathematical modeling is not only limited to dealing with one particular task, but they have to apply it to different situations that can be modeled by using a specific mathematical concept, relationships, or formula, developing a routine and fluency in mapping problem data to the basic mathematical model and in working through this model to reach a solution (Van Dooren, Verschaffel, Greer & De Bock, 2006). The modeling process has a cyclic nature where students do not move in succession through the different steps of the modeling process. The students, when modeling, go through many modeling cycles as they need to revisit their work many times, and gradually refine their model or sometimes reject it.
Figure 3: Schematic diagram of the process of modeling
In the literature, many researchers in their work on problem solving and modeling have presented diagrams for visualizing contextual mathematical problem- solving processes. Mathematical modeling of problem-solving is a complex procedure consisting of different stages (Van Dooren, Verschaffel, Greer & De Bock, 2006).
Verschaffel et al. (2000, p.xii) presented the Schematic diagram of the modeling process to represent the stages of the problem-solving procedure as shown in Figure 3.
At the first stage, students need to understand the phenomenon under investigation to create a model of the relevant elements, and relations rooted in the situation. Students need to identify the key and less important elements that should be included in the situation model. The second stage is building a mathematical model of the related items by mathematising the situation model. The situation model is mathematised by translating it into a mathematical equation involving the key quantities and relations. Then students obtain the solution by manipulating the model.
It is not enough just to get the answer, students need to evaluate their results against a situation model in which the students check their results are reasonable and appropriate to the original situation. At the final step, students are supposed to communicate the interpreted results considering the circumstances of the problem (Verschaffel et al., 2000).
Many Mathematical competencies, such as reading and communication, designing and applying problem solving strategies, or working mathematically (reasoning, calculating...) are closely related to modeling (Niss, 2010). By modeling, mathematics becomes more meaningful for learners and useful for cognitive analyses.
Blum and Leiß (2007) presented a modeling cycle for solving these tasks in the seven- step model. Mathematical modeling is seen as a cognitively challenging activity due
to the many competencies involved (Blum, 2015). This model presented by Blum (2015, p. 76) is shown in Figure 4 below.
Many students remain stuck in the first step which is understanding the situation and constructing the situation model. The modeling process begins with a problem from a real-life context and then ends again in a real model of the original situation after simplifying the real situation. Understanding the situation model in the cycle is a very important phase during the modeling process. This is because it describes the transition between the real situation and situation model as a phase of understanding the problem. Then during the process of mathematisation, this real model is transformed into a mathematical model. Then working mathematically leads to the mathematical results to finally interpret it in the real world as real results (Blum, 2015; Blum & Niss, 2010).
Figure 4: The seven-step modelling Schema
There are two types of mathematisation, horizontal and vertical arithmetic formulated by Treffers (1978). Vertical mathematics appears to mean “formal”
mathematics whereas horizontal mathematisation denotes the “informal” ML. The horizontal mathematisation is explained by (Freudenthal, 1991) as “going from the world of life into the world of symbols, while vertical mathematisation means moving within the world of symbols” (p. 24). Mathematisation is a term used by OECD which involves five elements describing how to solve a problem with roots in reality (Hope, 2007).
It is evident from the above discussion that the distinction between different concepts such as mathematisation and mathematical modeling is unclear. However, Blum and Niss (2010) clarified the difference as mathematisation is a one-way process that translates the real model into a mathematical model, whereas mathematical modeling is the process of translation between the real world and mathematics in both directions that involves the entire process.
Mathematical modeling has been considered as a cornerstone of the PISA framework for mathematics by OECD where it is incorporated into the definition of mathematical literacy that investigates the ability to deal with real-life contexts.
Students implement mathematics and use mathematics tools to solve contextual problems through a series of stages. The definition of mathematical literacy mentioned three processes describing what individuals do to relate the context of a problem to mathematics and thus solve the problem (OECD, 2013).
According to Stacey (2011), mathematical modeling consists of three processes: formulating, solving, and interpreting. Likewise, Brown and Schäfer (2006) describe the same cyclical processes using the terms formulation, analysis,
interpretation, and consolidation. These processes will be found in the central work of the teacher leading students from real-life situations to the application of appropriate mathematics. Moreover, these processes are key components of mathematical modeling and mathematical literacy as defined for cycle 2012 as well (OECD, 2013).
This modeling cycle is presented by OECD (2013, p. 26) is shown in Figure 5.
The mathematical modeling cycle takes place with a “problem in context”. To begin solving the contextual problem, the individual attempts to formulate the situation mathematically based on the relevant mathematics identified in the problem situation.
In this stage, the problem solver transforms the “problem in context” into a
“mathematical problem” to apply the mathematical treatment. Then, mathematical concepts, procedures, facts, and tools are employed to find “mathematical results”.
This stage where mathematical reasoning, manipulation, transformation, and computation take place. In the next stage, the “mathematical results” need to be interpreted in terms of the original problem as “results in context”. The problem solver needs to “interpret, apply, and evaluate” the mathematical solution in the real-world context of the problem (OECD, 2013).
Figure 5: The mathematical modeling cycle of PISA 2012 framework
In a modeling cycle, it is often not necessary to go through all its stages despite it being an essential aspect of the PISA conception of students as active problem solvers, especially in the context of an assessment (Niss et al., 2007). Students’ scores in the PISA 2012 indicated overall student achievement in the three processes based on the definition of mathematical literacy (OECD, 2013). However, the definition of mathematical literacy as “an individual’s capacity to reason mathematically and to formulate, employ, and interpret mathematics to solve problems in a variety of real- world contexts” (OECD, 2018) does not focus solely on problem-solving to be mathematically literate, but also put reasoning at the center of the problem-solving cycle. Figure 6 illustrates the relationship that links problem solving and reasoning in the modeling cycle of mathematics in the PISA 2021framework.
OECD (2018a, 2018b) stated that mathematically literate students could apply their mathematical knowledge to extract the abstract mathematical of the problem (more specifically contextual real-life problems) and then formulate it mathematically using appropriate terminology. This transformation process entails mathematical
Figure 6: The mathematical modeling cycle of PISA 2021 framework
reasoning. Next, they need to use mathematical concepts, algorithms and procedures taught in schools to solve the resulting mathematical problem. However, making the appropriate selection of those tools may require making a strategic decision that also demonstrates mathematical reasoning. Mathematical reasoning is also embedded in the process of evaluating and interpreting a solution within the original real-world situation (OECD, 2018a, 2018b).
There is an intersection between mathematical reasoning and solving real- world problems. In addition, mathematical reasoning goes beyond solving practical problems as it is also a way of evaluating and interpreting the quantitative nature of problem-solution that is best understood mathematically. Thus, mathematical literacy is seen to be a composite of two connected aspects that are problem solving and mathematical reasoning (OECD, 2018b). The PISA 2021 framework places mathematical reasoning at the heart of the problem-solving process. Based on the importance of mathematical reasoning for mathematical literacy, the next section will be devoted to studying mathematical literacy (OECD, 2018a, 2018b).